a: \(\sin^4\alpha+cos^4\alpha\)
\(=\left(\sin^2\alpha+cos^2\alpha\right)^2-2\cdot\sin^2\alpha\cdot cos^2\alpha\)
\(=1-2\cdot\sin^2\alpha\cdot cos^2\alpha\)
b: \(\sin^4\alpha-cos^4\alpha\)
\(=\left(\sin^2\alpha-cos^2\alpha\right)\left(\sin^2\alpha+cos^2\alpha\right)\)
\(=\sin^2\alpha-cos^2\alpha=1-cos^2\alpha-cos^2\alpha=1-2\cdot cos^2\alpha\)
c: \(\tan^2\alpha-\sin^2\alpha\)
\(=\frac{\sin^2\alpha}{cos^2\alpha}-\sin^2\alpha=\sin^2\alpha\left(\frac{1}{cos^2\alpha}-1\right)\)
\(=\frac{\sin^2\alpha\cdot\left(1-cos^2\alpha\right)}{cos^2\alpha}=\sin^2\alpha\cdot\tan^2\alpha\)
d: \(\sin^6\alpha+cos^6\alpha\)
\(=\left(\sin^2\alpha+cos^2\alpha\right)^3-3\cdot\sin^2\alpha\cdot cos^2\alpha\cdot\left(\sin^2\alpha+cos^2\alpha\right)\)
\(=1-3\cdot\sin^2\alpha\cdot cos^2\alpha\)
